# Randomised regression

January 14, 2017 — December 1, 2020

Tackling your regression, by using random embeddings of the predictors (and/or predictions?). Usually this means using low-dimensional projections, to reduce the dimensionality of a high dimensional regression. In this case it is not far from compressed sensing, except in how we handle noise. In this linear model case, this is of course random linear algebra, and may be a randomised matrix factorisation. You can do it the other way and prject something into a higher dimensional space, which is a popular trick for kernel approximation.

I am especially interested in seeing how this might be useful for dependent data, especially time series.

Brian McWilliams, Gabriel Krummenacher and Mario Lučić, Randomized Linear Regression: A brief overview and recent results. Gabriel implemented some of the algorithms mentioned, e.g.

• Subsampled Randomized Fourier Transform.
• SRHT: Uses the Subsampled Randomized Hadamard Transform (SRHT), equivalent to leverage based sampling.
• aIWS, aRWS: Samples based on approximated statistical influence.
• Uluru: SRHT with bias correction.

In the modern era of high-dimensional data, the interface between mathematical statistics and optimization has become an increasingly vibrant area of research. In this course, we provide some vignettes into this interface, including the following topics:

• Dimensionality reduction via random projection. The naive idea of projecting high-dimensional data to a randomly chosen low-dimensional space is remarkably effective. We discuss the classical Johnson-Lindenstrauss lemma, as well as various modern variants that provide computationally-efficient embeddings with strong guarantees.

• When is it possible to quickly obtain approximate solutions of large-scale convex programs? In practice, methods based on randomized projection can work very well, and arguments based on convex analysis and concentration of measure provide a rigorous underpinning to these observations.

• Optimization problems with some form of nonconvexity arise frequently in statistical settings – for instance, in problems with latent variables, combinatorial constraints, or rank constraints. Nonconvex programs are known to be intractable in a complexity-theoretic sense, but the random ensembles arising in statistics are not adversarially constructed. Under what conditions is it possible to make rigorous guarantees about the behavior of simple iterative algorithms for such problems? We develop some general theory for addressing these questions, exploiting tools from both optimization theory and empirical process theory.

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